Do we have to offer paid vacation? And if we do offer it, do we have to offer it to all employees?

In Illinois there is no requirement to offer paid vacation. If you do offer paid vacation, the rationale for the structure must be non-discriminatory.
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In Illinois there is no requirement that employers offer paid vacation.  However, many employers choose to offer some amount of paid vacation to some or all of their employees.  How you structure your paid vacation program is up to your company as long as the rationale is non-discriminatory.  You can, for instance, offer paid vacation based on full time versus part-time status, or seniority or type of position (exempt versus hourly).